Two thin convex lenses L1 and L2 of focal lengths f1 and f2, respectively, are placed coaxially in contact. An object is placed at a point beyond the focus of lens L1. Draw a ray diagram to show the image formation by the combination and hence derive the expression for the focal length of the combined system.
Draw a ray diagram to show the image formation by a combination of two thin convex lenses in contact. Obtain the expression for the power of this combination in terms of the focal lengths of the lenses.
Consider two thin lens L1 and L2 of focal length f1 and f2 held coaxially in contact with each other. Let P be the point where the optical centres of the lenses coincide (lenses being thin).
Let the object be placed at a point O beyond the focus of lens L1 such that OP = u (object distance). Lens L1 alone forms the image at I1 where P I1 = v1 (image distance). The image I1 would serve as a virtual object for lens L2 which forms a final image I at distance PI = v. The ray diagram showing the image formation by the combination of these two thin convex lenses will be as shown below:
From the lens formula, for the image I1 formed by the lens L1, we have
For the image formation by the second lens, L2
Adding (1) and (2) we get:
If the two lenses are considered a single lens of focal length f, which forms an image I at a distance v with an object distance being u, then we get
`1/v-1/u=1/f """ (where"1/f=1/f_1+1/f_2)`
Hence, the focal length of the combined system is given by `f=(f_1f_2)/(f_1+f_2)`
In terms of power, equation can be written as
`P = 1/f_1 + 1/f_2 (as P = 1/f)`